#### Vol. 9, No. 8, 2015

 Recent Issues
 The Journal Cover Editorial Board Editors' Addresses Editors' Interests About the Journal Scientific Advantages Submission Guidelines Submission Form Subscriptions Editorial Login Contacts Author Index To Appear ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print)
Indicators of Tambara–Yamagami categories and Gauss sums

### Tathagata Basak and Ryan Johnson

Vol. 9 (2015), No. 8, 1793–1823
DOI: 10.2140/ant.2015.9.1793
##### Abstract

We prove that the higher Frobenius–Schur indicators, introduced by Ng and Schauenburg, give a strong-enough invariant to distinguish between any two Tambara–Yamagami fusion categories. Our proofs are based on computation of the higher indicators in terms of Gauss sums for certain quadratic forms on finite abelian groups and rely on the classification of quadratic forms on finite abelian groups, due to Wall.

As a corollary to our work, we show that the state-sum invariants of a Tambara–Yamagami category determine the category as long as we restrict to Tambara–Yamagami categories coming from groups $G$ whose order is not a power of $2$. Turaev and Vainerman proved this result under the assumption that $G$ has odd order, and they conjectured that a similar result should hold for groups of even order. We also give an example to show that the assumption that $|G|$ is not a power of 2 cannot be completely relaxed.

##### Keywords
fusion category, Tambara–Yamagami category, Frobenius–Schur indicator, discriminant form, quadratic form, Gauss sum
##### Mathematical Subject Classification 2010
Primary: 18D10
Secondary: 15A63, 11L05, 57M27